Spectral analysis of the Moore-Penrose inverse of a large dimensional sample covariance matrix

JOURNAL OF MULTIVARIATE ANALYSIS

卷:148

Article

Bodnar, Taras¹  Dette, Holger²  Parolya, Nestor³ 

[1] Stockholm Univ, Dept Math, SE-10691 Stockholm, Sweden

[2] Ruhr Univ Bochum, Dept Math, D-44870 Bochum, Germany

[3] Leibniz Univ Hannover, Inst Empir Econ, D-30167 Hannover, Germany

关键词: CLT;    Large-dimensional asymptotics;    Moore-Penrose inverse;    Random matrix theory;   

DOI  :  10.1016/j.jmva.2016.03.001

来源: Elsevier

【 摘 要 】

For a sample of n independent identically distributed p-dimensional centered random vectors with covariance matrix Sigma(n) let (S) over tilde (n) denote the usual sample covariance (centered by the mean) and S-n the non-centered sample covariance matrix (i.e. the matrix of second moment estimates), where p > n. In this paper, we provide the limiting spectral distribution and central limit theorem for linear spectral statistics of the Moore-Penrose inverse of S-n and (S) over tilde (n). We consider the large dimensional asymptotics when the number of variables p -> infinity and the sample size n -> infinity such that p/n -> c is an element of (1, +infinity). We present a Marchenko-Pastur law for both types of matrices, which shows that the limiting spectral distributions for both sample covariance matrices are the same. On the other hand, we demonstrate that the asymptotic distribution of linear spectral statistics of the Moore-Penrose inverse of (S) over tilde (n) differs in the mean from that of S5. (c) 2016 Elsevier Inc. All rights reserved.

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